rainbow gravity and the very early universe yi ling center for gravity and relativistic astrophysics...

Rainbow Gravity and the Very Early Universe

Yi Ling Center for Gravity and Relativistic AstrophysicsNanchang University, ChinaNov. 4, 2006

2006 Workshop on Dark Universe, Suzhou

Contents

• Introduction: the semi-classical effect of quantum gravity and cosmology

• From deformed special relativity to gravity’s rainbow

• Modified FRW universe from rainbow gravity

• Two fundamental issues in quantum gravity

1. Dynamical problem

Gravity is a constrained system,

Diffeomorphism invariance ---observables

Hamiltonian constraint----problem of time

2. Classical limit of quantum gravity

The reconstruction of classical spacetime

Some problems related to loop quantum gravity

• Special relativity: Lorentz symmetry

• Testable signature of quantum gravity ： Gamma Ray Burst (2007 ）

• Particle physics ： Ultra high energy cosmic rays

• Quantum cosmology ： Big bang ， Inflation

• Classical limit ： Coherent states ， Gravitons

• Black Hole Physics: Black hole remnant and information Paradox

The fate of Lorentz symmetry at Planck scale

• Threshold anomaly of ultra high energy cosmic rays :

GZK cutoff

S.Coleman and S.Glashow, Phys. Rev.D 59, 116008 (1999)

J. Magueijo and L. Smolin, Phys. Rev. Lett(88) 190403,2002.

Modified dispersion relations

11 200 10 10thE Gev ev

2 2 20 ( , )E m p E p

CMBp p

CMBp p

2pth

p GZK

m mE E

E

2 2 20E m p 2 2 2

0 ( , )E m p E p

2 4

pp p

pthp

m

m m mE

E E

410E ev 17 2 2 20 210 ( ) (10 )ev E ev

2110 10thp GZKE ev E

Modified dispersion relations as semi-classical effect of quantum gravity

Quantum mechanics General relativity

Planck Length:

Planck Mass:

3 34: / 10pl G c m

5 19/ 1/ 10p pm c G l Gev

G

2 2 4 2 2 2 2( )npE m c p c l p p c

41, | | 10n

1, 8 1/ 1/p p plc l G m E

Three ways to obtain modified dispersion relations in semi-classcial approach to LQG

• The action of the Hamiltonian on the weave states in loop repr

esentation R.Gambini and J.Pullin, Phys. Rev. D 59, 124021 (1999)

• Kodama state in loop quantum gravity with a positive cosmological constant

L.Smolin, arXiv:hep-th/0209079.

• Coherent states for quantum gravity

H.Sahlmann and T.Thiemann, Class. Quant. Grav. 23, 909 (2006). arXiv:gr-qc/0207031.

The fate of Lorentz symmetry at Planck scale

• Modified dispersion relations from semiclassical LQG:

2 (1 )p i ids dt dt l e e

2 2 2 3 ...pE p m l E

0( , ) ( ) ( , )A A A

ia iaE he 0

0

ˆ ˆ( , ) ( ) ( , , )

(1 ) ( ) ( , , )

ia ia

iap

E A E A T a

E l A T a

22 2( )

(1 )i

p

km g k k

l

.0( ) ( ) ( , ) 0grav matterH H A A

Doubly Special Relativity(DSR)

• The relativity of inertial frames, two universal constant:

1) In the limit , the speed of a photon goes to a

universal constant, .

2) in the above condition is also a universal constant.

/ 0plE E

plE

c

2 2 2 2 21 2 0/ , / ,pl plE f E E p f E E m

As a result, the invariant of energy and momentum is modified to

1pl pE l

Non-linear Lorentz transformation in DSR

• In momentum space

00

0

0

0

0

0

( )'

1 ( 1)

( )'

1 ( 1)

'1 ( 1)

'1 ( 1)

z

p p z

zz

p p z

xx

p p z

yy

p p z

p vpp

l p l vp

p vpp

l p l vp

pp

l p l vp

pp

l p l vp

Two key issues in DSR

• The definition of position space

• Soccer problem

F

nonL

'p

px

1F'x

?

2 2 2 20

0 0

2 2 2 2 2 2 20 0

( )

,

( ) '( )

np

n n np p

E M P El E

M Nm E N P Np

m p N l m p l

?px

• Gravity’s rainbow

Non-linear map on momentum space:

To keep the contraction between position and momentum linear

The dual space is endowed with an energy dependent quadratic invariant.

2 2 2 2 21 2 0, ,E f E p f E m

iidx dp dtdE dx dp

2 2 22 2

1 2

1 1

( , ) ( , )ds dt dx

f E f E

0, , ( / ) , ( / )i i pl pl iU E p U U f E E E g E E p

From doubly special relativity to gravity’s rainbow or deformed general relativity

• Rainbow gravity as an extension of DSR into a general relativity:

1) Correspondence principle

2) Modified equivalence principle

Freely falling observers who take measurement with energy E, will observe the laws of physics to be the same as modified special relativity.

0

lim ( )

pl

ab abE

E

g E g

0 0

1 1;

/ /i i

pl pl

e e e ef E E g E E

1/ plR E E

J. Magueijo and L. Smolin, Class.Quant.Grav.21, 1725 (2004). arXiv:gr-qc/0305055

Rainbow metric and rainbow Einstein equations

• Rainbow metric

• Rainbow Einstein equation

0 0

1 1;

/ /i i

pl pl

e e e ef E E g E E

( ) aba bg E e e

( ) 8 ( ) ( ) ( )G E G E T E g E

Modified FRW universe

• The modified metric

22 2

2 21 2

1

( ) ( )i j

ij

a tds dt dx dx

f E f E

22

22 2

1 1

21

8 ( )( )

3 3

4 3 ( )( )

3 3

3 0

fa K EG E

a f a f

a p EG E

a f

ap p

a

( ) 8 ( ) ( ) ( )G E G E T E g E


• Generalization:

22 ( ) 1f E

The probe is identified with the radiation particle

( )E E t

( ) ; 0; 0; ij ijG E G K

Ansatz:

Modified FRW universe • The connection components:

• Non-trivial components of Riemann tensor:

• Ricci tensor components:

0 0 200 0, , i i

ij ij j j

f af aa

f a

00

0 20

2

i i ij j j

i j ij ij

i i ijkm k jm m jk

a afR

a af

R f aa ffaa

R f a

00

2 2

22

2

3 3

2

6

ij ij

a afR

a af

R f aa a ffaa

a af aR f

a af a


• Energy-momentum tensor:

• Unit vector:

• Conservation equation:

0

3 0

E T

H P

( )T u u P g u u

1( ,0,0,0) 1u f u u g


• Generalization of the modified FRW equations:

22

2

8

3

4

3 0

GH

f

P fH G H

f f

H P P


• Specify the function:

2 2 21 pf l E

2 2 4 2pE l E p

2 2 22

11 1 4

2 pp

E l pl

max

11/(2 ),

2p

p

p l El


• The averaged effect:

2

2 2

3 1

8

3 1 p

a

a

a G

a l E

2 2 21 pf l E


• The averaged effect:

• Define

2

2

1pl xx C

x

2 , 4a

Ea

2px l


• The solution:

2

1

t

2 1 11lnp

xl xt

C x x

0pl

0t 4

2 4

1p

p p

l

l Ct l


• The modified equations from LQG:

2 81

3

4 1 2

c

c

GH

H G P

12 4 41 pf l E

Remark 1: Statistics of photons with modified dispersion relations

• The state density

2

3

4( ) 2

V p dpg d

h

2 22 2 2 2 2

20

1 ( )p

p cE E p f c

l E

E

22

3 3 3 3

4 8 (1 '/ )( ) 2

V p dp V f fg d d

h h c f

Remark 1: Statistics of photons with modified dispersion relations

• Modified Stefan-Boltzmann law

2

24/0

1 4 100( ) 1

1 21 pkT

Ug d T l kT

V V e c

/ 21ln(1 ) ( ) (1 )

3kTPV

e g d P TkT

3 2/0

1( ) 1

1kTN g d VT T

e

2(1 )U

E kT TN

25.88

28.17

Remark 2:

• Impact on black hole physics

the modified black holes will not evaporate totally, but have a remnant which can be viewed as a candidate for dark matter.

• Unruh effect

Summary and Conjectures

• Rainbow gravity or deformed general relativity can be viewed as an effective theory at the semi-classical limit of quantum gravity. In this framework there is no single or fixed background spacetime, it depends on the energy of probes or test particles.

• We have considered some possible impacts on the FRW universe as well as Unruh effect and black holes in a heuristic way. More strict mathematical consideration is needed.

• The scheme presented here can be generalized to other sorts of modified black hole solutions.

• Conjectures 1)Tunneling effect 2) MOND

THANK YOU!

rainbow gravity and the very early universe yi ling center for gravity and relativistic astrophysics...

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